# Questions tagged [cauchy-product]

For questions about the Cauchy product, the discrete convolution of two sequences.

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### Why is the Cauchy product of the series $a_{n}:=b_{n}:=\dfrac{(-1)^n}{\sqrt{n+1}}$ diverging?

I'm new to series and I'm reading old questions here to see some examples of tests for divergence and convergence. In the following, I have a question about the conclusion: https://math.stackexchange....
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### Conditions for the Following Series to Converge

The problem is to show on which conditions the following series converge $$\sum_{n=0}^\infty\left({\sum_{k=1}^{n}a^k}{(-1)^{(n-k)+1}\frac{(n-k)^2}{e^{n-k}}} \right)$$ I tried to simplify the series ...
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### Showing that $2C(x)^2 = C(2x)+1$ using Cauchy product Formula

I'm stuck trying to calculate $2C(x)^2 = C(2x)+1$ with $C(x) =\sum_{n=0}^{\infty} (-1)^n \frac {x^{2n}} {(2n)!}$ (Cosine Power series) So I've seen proofs for this identity before, but never using the ...
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### How to prove this series Cauchy product?

Let the first series be $\sum_{k=0}^{\infty} u_{k}(x-a)^{k}$. Let the second series be $\sum_{k=0}^{\infty} v_{k}(x-a)^{k}$ Their convergence areas are $R_{1}$ and $R_{2}$. I don't know how to ...
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### Cauchy product summation converges

I had a previous question here, which I'm quoting: How can I prove that the following summation converges? $$\sum_{n=0}^\infty \sum_{k=0}^n \frac{(-1)^n}{(k+1) (n-k+1)}$$ I tried to prove ...
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### Cauchy Product of two geometric series.

So I wanted to calculate/prove that $$\frac{1}{(1-q)^2} = \displaystyle\sum_{n=0}^{\infty}(n+1)q^n,$$ $q\in \mathbb{C},$ $\mid q \mid < 1$ using the Cauchy-Product. So this is my solution and I'm ...
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### show the power series converges for |x|<1, and find the closed forms

\begin{matrix} 1) S(x) = \sum_{n=0}^\infty (1+ \frac{1}{1!} + \frac{1}{2!} + ...+ \frac{1}{n!}) x^n \\ 2) S(x) =\sum_{n=1}^\infty a_n x^n, a_n= \Big\{ \begin{alignedat}{3} -2/n ,n=3k \\ 1/n,n \neq 3k ...
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I have a mistake that I can't find somewhere along the way. Please help me find the place things go wrong. Find the product of $\cos x$ and $\sin x$ as defined: $$\cos(x) = \sum_{k=0}^{\infty} \frac{\... • 161 0 votes 0 answers 72 views ### Cauchy product of 2 series Let z \in \mathbb{C}. I want to prove that$$\sum_{n\in\mathbb{Z}} \frac{(-1)^n}{z-n} \sum_{n\in\mathbb{Z}} \frac{(-1)^n}{z-n} = \sum_{n\in\mathbb{Z}} \frac{1}{(z-n)^2}.$$Using the Cauchy Product ... • 160 0 votes 1 answer 53 views ### How to show that these sums are equal? Assuming that the product is associative, I would like to show that$$ \sum_{s=0}^{k} (\sum_{i=0}^{s} (\alpha_i \cdot \beta_{s-i}) \cdot \gamma_{k-s}) = \sum_{s=0}^{k} ( \sum_{i=0}^{k-s} \alpha_s \...
I have the following recurrence relation that I am trying to solve: $a_0 = 1$ $$a_n = \sum_{i = 0}^{n-1} (i+1)a_i$$ for $n \geq 1$. Let's define $A(x) = \sum_{n= 0}^\infty a_nx^n$, so multiplying ...